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Evolutionary dynamics at the tumor edge revealmetabolic imaging biomarkersJuan Jimenez-Sancheza,1 , Jesus J. Bosquea,1 , German A. Jimenez Londonob, David Molina-Garcıaa ,Alvaro Martıneza,c , Julian Perez-Betetaa , Carmen Ortega-Sabatera, Antonio F. Honguero Martınezd,Ana M. Garcıa Vicenteb, Gabriel F. Calvoa,2,3 , and Vıctor M. Perez-Garcıaa,2,3

aMathematical Oncology Laboratory, Universidad de Castilla-La Mancha, Ciudad Real, 13071, Spain; bNuclear Medicine Unit, Hospital General Universitariode Ciudad Real, Ciudad Real, 13005, Spain; cDepartment of Mathematics, Universidad de Cadiz, Cadiz, 11009, Spain; and dThoracic Surgery Unit, HospitalGeneral Universitario de Albacete, Albacete, 02006, Spain

Edited by Michael E. Phelps, University of California, Los Angeles School of Medicine, Los Angeles, CA, and approved January 4, 2021 (received for reviewAugust 26, 2020)

Human cancers are biologically and morphologically heteroge-neous. A variety of clonal populations emerge within theseneoplasms and their interaction leads to complex spatiotempo-ral dynamics during tumor growth. We studied the reshapingof metabolic activity in human cancers by means of continuousand discrete mathematical models and matched the results topositron emission tomography (PET) imaging data. Our modelsrevealed that the location of increasingly active proliferative cel-lular spots progressively drifted from the center of the tumorto the periphery, as a result of the competition between gradu-ally more aggressive phenotypes. This computational finding ledto the development of a metric, normalized distance from 18F-fluorodeoxyglucose (18F-FDG) hotspot to centroid (NHOC), basedon the separation from the location of the activity (prolifera-tion) hotspot to the tumor centroid. The NHOC metric can becomputed for patients using 18F-FDG PET–computed tomography(PET/CT) images where the voxel of maximum uptake (standard-ized uptake value [SUV]max) is taken as the activity hotspot. Twodatasets of 18F-FDG PET/CT images were collected, one from 61breast cancer patients and another from 161 non–small-cell lungcancer patients. In both cohorts, survival analyses were carriedout for the NHOC and for other classical PET/CT-based biomarkers,finding that the former had a high prognostic value, outperform-ing the latter. In summary, our work offers additional insights intothe evolutionary mechanisms behind tumor progression, providesa different PET/CT-based biomarker, and reveals that an activityhotspot closer to the tumor periphery is associated to a worstpatient outcome.

cancer | 18F-FDG PET /CT | evolutionary dynamics | prognostic biomarker

Human cancers are genetically and morphologically heteroge-neous (1, 2). This is generally attributed to the evolutionary

dynamics of different clonal cell populations coexisting in thetumor ecosystem and undergoing stochastic branching processesover time (3–5). Successively acquired driver mutations, somaticalterations, and nongenetic modifications may confer increasedfitness on certain cancer cell phenotypes, which subsequentlyoutcompete those that do not experience such selection benefitswithin their microenvironment (4, 6, 7). Cells with specific advan-tageous traits may not show uniform spatial distribution acrossthe tumor, particularly in large tumors. In fact, trade-offs existthat preclude the occurrence of optimal phenotypes, as exempli-fied by the hallmarks of cancer (8), and thus only local selectionis expected to take place. This produces the spatial phenotypicdiversity found in primary tumors and distant metastases (9).

Sustained metabolic reorganization during tumor progression,due to bioenergetically very demanding processes such as rapidproliferation, is a major hallmark of cancer (8, 10). This gives riseto a global metabolic plasticity and fitness optimization that con-fers evolutionary advantages under specific selective pressures,such as hypoxia (11). Positron emission tomography (PET) hasbeen proposed as a way to assess macroscopic tumor hetero-

geneity in human patients (12). The technique is used in clinicalpractice with the radiotracer 18F-fluorodeoxyglucose (18F-FDG)(13), which is an analog of glucose and thus a marker of glycoly-sis (14). The altered tumor metabolism leads to an up-regulationof glycolysis and an increase in glucose consumption (15). Thishappens even in the presence of oxygen and is referred to as theWarburg effect. Even though this process is energetically ineffi-cient (16), cancer cells may find it beneficial to satisfy the biomassdemands required by their high proliferation rates (17). This isconfirmed by studies that relate the uptake of 18F-FDG in PETimages to proliferation markers (18, 19). Therefore, the spatialmap of glucose consumption provided by 18F-FDG PET images,as measured at each voxel by the standardized uptake value(SUV), is of great utility in portraying the spatial distribution ofproliferation within the tumor.

The degree and impact of intertumor diversity and intratumorheterogeneity in patients has driven the need for quantitativeframeworks to account for this variability (20). We consid-ered how the metabolic activity might be distributed inside thetumor and how that information could be related to 18F-FDG

Significance

Through the use of different in silico modeling approachescapturing tumor heterogeneity, we postulated that areas ofhigh metabolic activity would shift toward the periphery astumors become more aggressive. To confirm the hypothesisand provide clinical value for the finding, we collected 18F-FDGPET images of breast cancers and non–small-cell lung cancers,where we measured the distance from the point of maximumactivity to the tumor centroid, normalizing it by a surrogateof the volume. The metric, NHOC, showed higher prognos-tic value than other classical PET-based metabolic biomarkersused in oncology, evidencing that the shift of the hotspotof activity from the center of the tumor to its peripherycorrelates with a poor prognosis.

Author contributions: G.F.C. and V.M.P.-G. designed research; J.J.-S., J.J.B., D.M.-G.,A.M., C.O.-S., and G.F.C. performed research; J.J.-S., J.J.B., G.A.J.L., D.M.-G., A.M., J.P.-B.,A.F.H.M., A.M.G.V., G.F.C., and V.M.P.-G. analyzed data; J.J.-S., J.J.B., G.F.C., andV.M.P.-G. wrote the paper; and G.A.J.L., J.P.-B., A.F.H.M., A.M.G.V., and V.M.P.-G. collectedand processed the data and analyzed the medical implications.y

The authors declare no competing interest.y

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).y

This article is a PNAS Direct Submission.y1 J.J.-S. and J.J.B. contributed equally to this work.y2 G.F.C. and V.M.P.-G. contributed equally to this work.y3 To whom correspondence may be addressed. Email: Gabriel.Fernandez@uclm.es orvictor.perezgarcia@uclm.es.y

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2018110118/-/DCSupplemental.y

Published February 3, 2021.

PNAS 2021 Vol. 118 No. 6 e2018110118 https://doi.org/10.1073/pnas.2018110118 | 1 of 9

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PET images. Specifically, we looked at whether the location ofprominent proliferation hotspots, as measured by the voxel ofmaximum radiotracer uptake (SUVmax), could convey informa-tion about patient prognosis. We did this by analyzing how thesespots changed over time and space within the tumor in silicousing two mathematical models of different levels of complexity.The implications of these results, embodied in the definition of aprognostic biomarker named NHOC, were tested on datasets ofbreast and lung cancer patients.

ResultsPhenotype Variability Supports a Drift of the Highest MetabolicActivity toward the Tumor Boundary. To describe the emergenceof metabolic heterogeneity, we studied in silico a simple biolog-ical scenario assuming the tumor to be composed of a clonalpopulation of cells that can migrate, proliferate until the physicalspace is full, and die. To account for phenotypic heterogeneity,a transition probability that a cell proliferating at a rate ρ couldincrease or decrease its rate was introduced. The mathematicalmodel used was a continuous nonlocal Fisher–Kolmogorov-typeequation (21) which considered the tumor cell population to bestructured both by a spatial position vector x∈Ω⊂R3, insidea domain Ω, and a proliferation rate ρ∈ [0, ρm], where ρm is amaximum proliferation rate. Let u = u(x, ρ, t) denote the celldensity function, such that u(x, ρ, t) d3x dρ represents the num-ber of tumor cells that, at time t , have a proliferation rate ρ atpoint x. We modeled the dynamics of u(x, ρ, t) via the followingmigration–proliferation integro-differential equation:

∂u

∂t=Dc∇2u +Dρ

∂2u

∂ρ2

+ (ρ−µ)

(1− 1

K

∫ ρm

0

u(x, ρ′, t) dρ′)u(x, ρ, t). [1]

The first term accounts for cell migration with a diffusionconstant Dc> 0. The second term captures the effect of non-genetic instability, mediated by fluctuations in the proliferation

phenotype occurring with a diffusion constant Dρ> 0. Note thatthe proliferation phenotype is a hallmark in tumors resultingfrom alterations in growth regulation (8). The third term com-prises two main factors. The first one includes the proliferationrate ρ minus a constant death rate µ> 0; those cells having alarger factor ρ−µ will tend to display a fitness advantage unlessexogenous mechanisms (e.g., cytotoxic drugs targeting activelydividing cells) exert a negative selection effect on the pheno-type. The second factor consists of a nonlocal logistic form witha carrying capacity K > 0. This factor represents the interplaybetween intratumor subpopulations with different proliferationscompeting for the available space.

A number of quantities are useful for summarizing the infor-mation contained in Eq. 1. The first one is the marginal celldensity n(x, t) =

∫ ρm0

u(x, ρ, t) dρ, with typical radially symmet-ric profiles as shown in Fig. 1A. The second one is the prolifer-ation density M(x, t) (Eq. 9 in Materials and Methods), whichgives the spatiotemporal proliferation map and allows the tumorregions with high metabolic activity to be identified. Fig. 1Bdepicts M(x, t)/K and shows how the location of the highestactivity shifts from the tumor centroid toward the boundary as itgrows in silico. This observed displacement, which was found tobe linear with time, was quantified using two metrics. The firstone was the distance from the highest activity, corresponding tothe point of maximum proliferation, to the tumor centroid. Wenamed this metric the distance from the metabolic hotspot to thetumor centroid (HOC). To make HOC independent of size, sothat it can be compared among different tumors, we conceiveda second metric, normalized HOC (NHOC), defined as the ratiobetween HOC and the mean metabolic radius of the tumor, Rmet

(Fig. 1C and Eq. 10 in Materials and Methods). Simulations of Eq.1 showed that, during the early stages of the natural history of thetumor, the metric HOC was found to be zero or very small. How-ever, as the inner regions were filled with cells, HOC increasedlinearly with time (Fig. 1D). NHOC changed steadily from zeroto one, since the maximum proliferation spot can only occurbetween the tumor center and its edge, indicating that this spot

A

B

C

D

Fig. 1. Nonlocal Fisher–Kolmogorov model 1 predicts a drift of the highest metabolic activity from the tumor centroid to the periphery with time. (A)Normalized cell density n(x, tj)/K at tj= 6, 12, 18, 24, 30, and 36 mo (from left to right) for a radially symmetric tumor. (B) Pseudocolor plots of thenormalized spatiotemporal proliferation densityM(x, t) and profiles (Inset) ofM(x, tj) calculated at tj= 6, 12, 18, 24, 30, and 36 mo. (C) Mean metabolicradius Rmet(t) and (Inset) average proliferation rate ρa(t). (D) Variation over time of the distance from the tumor centroid to the hotspot of proliferation(HOC) and (Inset) normalized HOC by the mean metabolic radius (NHOC). Simulation parameters are listed in Materials and Methods.

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will move from the central regions of the tumor to its boundaries(Fig. 1 D, Inset).

The simulations of Eq. 1 revealed other noteworthy effects.First, the amplitude of the maximum activity in M(x, t) grewwith time, meaning that tumors at later stages of their evolution-ary history show increasing hotspot activity values. This was inline with the frequently observed association between SUVmaxand prognosis for different tumor histologies (22, 23). Second,the distribution of the proliferation rates displayed sustainedgrowth toward higher values of ρ, reflected in the average pro-liferation rate ρa (Fig. 1 C, Inset). The growth of the tumor pro-liferation rate with time (size) has been experimentally observedin other studies (24).

Genotype Evolutionary Dynamics Support the Drift of Tumor High-est Metabolic Activity toward the Boundary. We next resorted toa more complex and realistic biological scenario accounting forgenotypic alterations. We did this by considering a stochas-tic discrete model based on fundamental cell features. At thecellular level, cancer cells can be characterized by four deregu-lated processes: proliferation, migration, mutation, and death.These processes can be easily implemented as rules in a dis-crete mathematical model to mimic the main characteristics ofthe real system, with the drawback of facing high computationalcost, especially when simulating clinically relevant volumes (25).To overcome this problem, we developed a hybrid stochasticmesoscale model of tumor growth that allowed clinically rele-vant tumor sizes to be simulated while retaining the basic cancerhallmarks (24).

The model was parameterized for two of the most promi-nent cancer types, namely breast and lung cancer (non–small-celllung carcinoma [NSCLC]). A summary of our available datacan be seen in Table 1. Mutational landscapes were constructedbased on a simplification of their known mutational spectra.Alterations in EGFR and ALK, which are strongly associatedwith nonsquamous lung adenocarcinoma, were considered tomodel NSCLC, while driver mutations in PIK3CA and TP53were considered for breast cancer (26–29). Therefore, the muta-tional tree in both types of tumors simulated had two possiblealtered genes, leading to four possible combinations or “geno-types” that define four different clonal populations. Basal ratesassociated a characteristic time to each basic cell process, andmutation weights determined how these basal rates were affectedonce a given alteration was acquired. Mutation weights weretaken to contribute equally for all alterations, and their effectwas cumulative, so that a cell carrying two alterations simul-taneously would perform basic processes with a double advan-tage. Thus, the stochastic mesoscopic model provided a richerscenario to explore intratumoral heterogeneity during tumorgrowth.

We ran 100 simulations of breast cancer and 100 simulationsof NSCLC with random parameters uniformly sampled from the

ranges in Tables 1 and 2 (Materials and Methods). Cell number,activity (number of newborn cells), and most abundant clonalpopulation were calculated for each voxel and time step (Fig. 2).Tumor volumes were measured from the number of voxels con-taining more than a threshold number of cells Nt (Table 1),and the mean spherical radii (MSR) were computed fromthese.

As cells mutated in silico, new clonal populations emergedwith higher, more advantageous migration and proliferationrates. These new clones increased their relative abundance inthe tumor, eventually becoming fixed in the system. As thetumors grew larger, cell division occurred preferentially at thetumor periphery. This was as expected, since inner voxels becameprogressively filled with cells that prevented them from prolifer-ating. Voxels where the most aggressive clonal population wasmore abundant were associated with hotspots of maximum pro-liferation. Therefore, evolution was pushed toward the tumoredge: Cells with higher fitness (especially those having higherproliferation rates) appeared farther from the tumor centeras they grew. At each time step, the maximum proliferationspot was identified as the voxel with the largest number of cellbirths, and its distance to the tumor centroid (HOC) was cal-culated. Fig. 2 D and E shows a monotonic increase of HOCwith time for both histologies. Normalizing with respect to theMSR to get the NHOC showed that the point of maximumproliferation was displaced toward the boundary (Fig. 2 F andG) in all of the simulations performed. The only differencebetween simulations was the time that the maximum prolifera-tion spot took to reach the edge. Thus, NHOC was predictedto be a robust property related to the evolutionary state of thedisease.

PET Imaging Data Confirm Evolutionary Dynamics of the MaximumMetabolic Activity Validating Related Biomarkers. The computa-tional results suggested that NHOC might contain meaningfulprognostic information. In the clinical setting, the metabolicactivity distribution of the tumor can be evaluated by meansof 18F-FDG PET, which reflects the biological processes takingplace at a lower level (30) and is frequently used on newly diag-nosed breast cancer and NSCLC patients. To confirm or refutethe theoretical predictions, we performed a study on our patientcohort (Materials and Methods). For each patient, the tumorwas delineated in the images and the locations of centroids andSUVmax were obtained computationally from the segmented dis-tribution, as detailed in Materials and Methods. The metabolictumor volume (MTV), total lesion glycolysis (TLG) (integral ofthe SUV distribution over the volume), and NHOC metrics werecalculated for all tumors for both histologies. Two typical exam-ples of 18F-FDG PET images from breast cancer patients areshown in Fig. 3 A and B, respectively. Small values of NHOC,with SUVmax close to the tumor centroid as in Fig. 3 B and I, wereexpected to correspond to less developed disease, in accordance

Table 1. Stochastic model parameters

Parameter Meaning Value Ref.

L No. of voxels per side 80 —∆x Voxel side length (mm) 1 (62)∆t Time-step length (h) 24 —K Carrying capacity per voxel (cells) 2 · 105 (63)Nt Threshold cell number 0.2 · KN0 Initial population (cells) 1 —Vend Maximum reachable tumor volume (cm3) 50 (breast)

120 (lung) (64)Vdiag Tumor volume at diagnosis (cm3) 0.3 to 5 (breast) (65)

0.2 to 15 (lung) (66)

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Table 2. Basal rates and mutation weights

Processes Proliferation Migration Death Mutation

Breast basal rates (d−1) 0.0133 to 0.04 0.02 to 0.0303 0.01 to 0.02 0.01 to 0.02TP53/PIK3CA (%) 20 to 40 25 to 45 (−30) to (−10) 25 to 40Lung basal rates (d−1) 0.04 to 0.2 0.03 to 0.2 0.03 to 0.1205 0.03 to 0.1205TP53/KRAS (%) 20 to 40 25 to 45 (−30) to (−10) 25 to 40

with the previous theoretical framework. In contrast, the casesshown in Fig. 3 C and J with SUVmax displaced in relation to thecentroid would correspond to tumors with a poorer prognosis.

The histograms in Fig. 3 D–G and K–N depict the distributionsof MTV, TLG, SUVmax, and NHOC for both histologies. It isnoteworthy that the NHOC has a more regular distribution thanthe other PET-based measures, with definite values between 0and 1 and a centered mean (breast cancer, 0.51± 0.18, median0.50; NSCLC, 0.43± 0.2, median 0.39). It is clear from Fig. 3G and N that at the time of diagnosis the point of maximumuptake is typically located away from the geometrical center ofthe tumor. To inspect the relationships between the PET vari-ables, we calculated a correlation matrix analysis (SI Appendix,Fig. S1). The results show that NHOC does not strongly correlatewith the conventional measures and can therefore be consideredas an independent metric.

The classical measures (MTV, TLG, SUVmax) are known to beprognostic biomarkers in breast cancer and NSCLC (22, 23). Forthese variables we performed Kaplan–Meier analyses on over-all survival (OS) and disease-free survival (DFS) (SI Appendix,Figs. S3 and S4). All of the variables had prognostic value in thebreast cancer cohort, but only MTV returned significant results(P value < 0.05) in the NSCLC cohort.

We then tested the prognostic value of NHOC by Kaplan–Meier analyses with OS and DFS as endpoints (Materials andMethods). Results for the best splitting thresholds are shownin Fig. 4. For the breast cancer cohort, NHOC showed robustresults in terms of OS, with a best splitting threshold in bothOS and DFS of NHOC = 0.499 (Fig. 4 A and C). Interestinglyfor OS, the most relevant metric, the C index, reached an out-standing value of 1 (for DFS it was 0.899). Thus, no patients withtumors having their SUVmax closer than half the radius (n = 30)died from the disease. In NSCLC, NHOC separated the patientswell, and the best splitting threshold, NHOC = 0.64, led to aC index of 0.875 for OS. The separation in median OS betweengroups was 57.33 mo, while in DFS it was 36.62 mo. Therefore,patients for whom the 18F-FDG hotspot displays an increasingshift from the tumor centroid are associated to a worst outcome.

To determine whether the prognostic value of NHOC mightbe related to possible necrotic regions, we evaluated the presenceof hypometabolic voxels inside the delineated tumor, i.e., thosesituated inside the tumor that have a lower SUV value than thesegmentation threshold. We found that these regions are presentonly in 7 of the 61 patients of our breast cancer cohort and in13 of the 161 patients of our NSCLC cohort. Moreover, tumorswhere hypometabolic voxels are detected account on average

Fig. 2. Hybrid stochastic mesoscale model shows that competition among progressively more aggressive phenotypes is pushed to the edge. (A) Three-dimensional volume renderings at different time frames (from left to right: 78, 85, and 92% of simulation) of a simulation of breast cancer growth depictingclonal populations within the tumor. Color of cell populations ranges from green (less aggressive) to red (more aggressive). Rates are proliferation 0.0315d−1, death 0.0157 d−1, mutation 0.0160 d−1, and migration 0.0235 d−1. (B) Central section for the same simulation and time frames as in A showing themost abundant clonal population per voxel. (C) Central section of tumor activity for the same time frames as in A. (D and E) HOC progression for everysimulation of breast cancer (D) and NSCLC (E). (F and G) Longitudinal NHOC dynamics for simulations of breast cancer (F) and NSCLC (G) growth, withindividual runs colored in reddish orange and all-simulation averaged NHOC in blue; crosses depict the time points at which each simulation ended.

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Fig. 3. Analysis of NHOC in 18F-FDG PET images reveals well-behaved distributions with prognostic potential. (A and H) Examples of PET images for breastcancer (A) and NSCLC (H) patients in our dataset. (B, C, I, and J) Two-dimensional slices from patients with small (B and I) and large (C and J) NHOC values forbreast cancer (B and C) and NSCLC (I and J) patients. The centroid of each segmented lesion and the voxel of SUVmax are marked with a white cross and agreen dot, respectively. (Scale bars, 1 cm.) (D–G and K–N) Histograms showing the distributions of metabolic tumor volume (D and K), total lesion glycolysis(E and L), SUVmax (F and M), and NHOC (G and N) for breast cancer (D–G) and NSCLC (K–N) patients in our datasets.

for only 1.63% of the total voxels in the breast cancer cohortand 2.68% of the voxels in the NSCLC cohort. Consequently,in our cohorts, the NHOC significance is not associated to theoccurrence of necrotic regions assessed by the manifestation ofhypometabolic voxels in the images.

The presence of lymph node metastases is a strong predictorof outcome in breast and lung cancers, with nodal metastaseshaving negative prognostic significance. We evaluated whetherthere was a significant difference in the NHOC value for thegroup of patients which showed nodal metastases on diagno-sis and those who did not (Mann–Whitney test; SI Appendix,Fig. S9). The P values were 0.4683 for the breast cancer datasetand 0.1071 for the NSCLC, thus discarding any significant dif-ference across groups and indicating that the information car-ried by NHOC is independent of the existence of lymph nodemetastases.

These results show the strength of NHOC as a prognosticbiomarker in comparison with the classical metrics. For OS in thebreast cancer cohort, only MTV approached the performance ofNHOC; however, the C index for NHOC (C index = 1) outper-formed the result for MTV (C index = 0.875). For DFS, noneof the classical variables showed nonisolated thresholds leadingto a statistically significant association between subgroups. Inthe lung cancer cohort, the NHOC metric outperformed, once

again, the prognostic value of the classical variables. RegardingOS, there were ranges of thresholds of MTV, TLG, and SUVmaxleading to statistically significant results with best C indexes of0.736 (MTV), 0.682 (TLG), and 0.658 (SUVmax) still substan-tially lower than the value obtained for NHOC (0.875). Resultsfor DFS were again similar, with only MTV and TLG achievingsignificance, with best values of 0.638 (MTV) and 0.607 (TLG),but still underperforming NHOC, with a C index of 0.651.

Discussion and ConclusionHeterogeneity is one of the hallmarks of tumor malignancy (1,2). Many mathematical models have been constructed account-ing for different aspects of the development of heterogeneitythrough evolutionary dynamical processes in a number of can-cer types (31–33). We did not intend here to develop a universalmathematical model to describe every aspect of tumor growthprogression, but rather to focus on understanding the basicdynamics of the hotspot of metabolic activity, due to the poten-tial applicability of the results. Different levels of complexitywere considered in each of the two complementary models con-structed, and both led to the same conclusions: 1) Tumors wouldevolve toward higher proliferation rate values, and 2) maximummetabolic activity would move toward the tumor edge as thetumor evolves with time.

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A

C

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D

Fig. 4. Kaplan–Meier curves obtained for best splitting thresholds corre-sponding to the NHOC metric. (A) Overall survival in breast cancer cohort.(B) Overall survival in NSCLC cohort. (C) Disease-free survival in breast cancercohort. (D) Disease-free survival in NSCLC cohort.

It is interesting to note that conclusion 2 (but not conclusion 1)would have been arrived at through the use of a “classical” localFisher–Kolmogorov model (34). In the context of that math-ematical model, proliferation is inhibited in areas of high celldensity and higher proliferation areas would switch from thetumor core to the periphery, as predicted also by the nonlocalmodel Eq. 1. This is, therefore, a robust finding of the study, andit is likely to be anticipated by other mathematical approaches.Conclusion 1 does not come as a surprise, since the maximummetabolic activity obtained from PET images (e.g., as measuredby the SUVmax) has been known to contain prognostic informa-tion in different cancer types (22, 23). Thus, the fact that SUVmaxpossesses a prognostic value in the clinical setting is compati-ble with the results of our models, where the activity grows withtime over the tumor’s natural history until a maximum value isreached.

Many studies have correlated either classical PET-derivedmetrics such as MTV and TLG (35, 36) or complex spatial fea-tures of the distribution of SUV values (37) with the outcome ofthe disease. However, no study has analyzed the prognostic valueof metrics derived from the location of the highest metabolicactivity. The fact that this simple biomarker has a high prognosticvalue is remarkable and probably related to the robustness of thebiological assumptions behind the mathematical models used tosubstantiate it. In fact, it is natural to expect that the presenceof more aggressive glucose-avid cells, that might be unable toprogress when located in saturated areas near the center of thetumor, may be a risk factor when placed in regions with muchmore capacity to settle and invade.

In our study we chose to take the voxel bearing the SUVmax asthe location of maximum metabolic activity to compute NHOC.We could also have used SUVpeak (the maximum SUV appearingin the distribution when all of the voxels are averaged with their26 neighbors), which is thought to be more stable and to betterdefine an extended region of high uptake (38). However, SUVmaxis often placed in the area defined by SUVpeak, thus leading toequivalent metrics (39). SUVmax is also easier to identify visually

and is therefore easier to use in clinical practice, besides beingthe simplest option.

The fact that NHOC provides an accessible and powerfulprognostic metric could be extended in different ways. First, itwould be valuable to look at whether changes in this biomarkermight provide a robust indication of an increase in malignancyfor initially indolent tumors (e.g., benign lung nodules, low-gradegliomas, etc.) undergoing a malignant transformation. Second,an intriguing open question would be to determine whetherthe rate at which NHOC changes during patient follow-up cor-relates with the occurrence and fixation of specific mutations.Finally, it may be the case that changes with time of this metric,after different treatment modalities, could help in assessing theresponse through sequential PET studies as a measure of howmuch NHOC is reduced.

Hotspots in staging 18F-FDG PET–computed tomography(PET/CT) have previously been reported as preferential sitesfor relapse after chemoradiotherapy in several types of cancer,including NSCLC (40, 41). Some authors have found that thesehotspots colocalize with hypoxic regions as inferred from HIF-1αtumor immunostaining (42) and are thus very relevant in the con-text of radioresistance (40, 43, 44). Furthermore, these hotspotshave also been linked to the occurrence of somatic mutations inlung cancer (45, 46) similarly to what is described in this work.Although further research is needed, this rationale has led someauthors to assert that patients may benefit from hotspot-baseddose escalation (43, 47). The voxel of SUVmax, hence, has a pre-eminent significance both in prognosis and in therapy and, as itis now shown here, its relative position has a direct bearing on itsrelevance. In addition to NHOC tracking throughout the disease,NHOC estimation in hypoxia-specific PET imaging, employingother radiotracers such as fluoromisonidazole (18F-MISO) or flu-oroazomycin arabinoside (18F-FAZA), could provide valuableinformation complementary to that of 18F-FDG PET/CT images.

Mathematical and computational models are progressivelygaining their place among the tools that are used to studycancer (48). In silico models based on evolutionary dynamicsmay capture relevant aspects of tumor growth and have provedhelpful in understanding tumor clonal heterogeneity, one ofthe main hallmarks of cancer (49). Mechanistic mathematicalmodels of different levels of complexity have been shown to pro-vide biomarkers of clinical significance (24, 50–57). This typeof approach provides a rational alternative to radiomic anddeep-learning studies, where a mechanistic explanation is oftenmissing. The study described in this paper falls into the formercategory, demonstrating that an informed understanding of thesystem’s emergent properties can shed light on the deeper rootsof its working.

It is worth mentioning that our mathematical approach,beyond its fundamental interest, has led to the proposal of asimple metric with high prognostic value that can be obtainedfrom 18F-FDG PET studies. The NHOC biomarker was ableto separate patients with breast cancer and non–small-cell lungcancer into two groups with significantly different survival (bothoverall and disease-free) and proved to be more powerful thantraditional 18F-FDG PET/CT biomarkers (MTV, TLG, SUVmax)currently used in clinics. This demonstrated that the geomet-ric location of the maximum metabolic activity, and not only itsvalue, contains information of clinical significance: Tumors inwhich the hotspot is located farther from the center are foundto have a worst prognosis.

This study opens many further avenues for research. The firstone is the search for other biomarker definitions accountingfor the location of highest metabolic activity. Second, it wouldbe interesting to test our findings in other tumor histologies,such as lymphoma or melanoma (58, 59). PET is a main-stream technique, increasingly employed in clinics and in many

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imaging studies for which a broad spectrum of tumor histolo-gies is available. Thus, the applicability of NHOC to other cancertypes would be an interesting extension of our work.

In conclusion, by using two mathematical models incorporat-ing evolutionary dynamics, we have shown that the maximummetabolic activity is expected to increase in magnitude andto move toward the tumor boundary as human solid tumorsprogress. On the basis of the theoretical predictions we havedefined a metric, the NHOC, representing the normalized dis-tance from the point of maximum activity to the tumor centroidand validated it as a prognostic biomarker in lung and breastcancer patients using PET imaging datasets. The biomarkerNHOC outperformed classical PET-based biomarkers such asTLG, MTV, and SUVmax and provides a notable example ofmathematically grounded research with applicability in oncology.

Materials and MethodsPatients. Our study was based on data from two different studies, bothof them approved by the Institutional Review Board (IRB) of Ciudad RealUniversity General Hospital (HGUCR). Breast cancer patients were partic-ipants of a multicenter prospective study, and written informed consentwas obtained from all patients. The inclusion criteria were 1) newly diag-nosed locally advanced breast cancer with clinical indication of neoadjuvantchemotherapy, 2) lesion uptake higher than background (i.e., those havinga SUVmax larger than twice the background activity readings), 3) absenceof distant metastases confirmed by other methods prior to the request ofPET/CT for staging, and 4) breast lesion size of at least 2 cm. Sixty-onepatients (18% lobular carcinoma, 82% ductal carcinomas, 100% women,age rank 25 to 80 y, median 50 y) were included in this dataset. TheTNM data were 54% T2, 18% T3, 28% T4; 28% N0, 55% N1, 6% N2,11% N3; 100% M0.

One hundred seventy-five patients (153 men, 22 women, age rank 41 to84 y, median 65 y) were included in the study from a dataset of lung cancerpatients who received surgery in the period June 2007 to December 2016.Histologies were 63 squamous-cell carcinomas and 112 adenocarcinomas.Staging information was 69 stage I, 70 stage II, 33 stage III, 3 stage IV. TheN staging was 107 patients N0, 46 N1, and 22 N2. All patients had M0. PETprotocol and machine were as in subgroup 1. The inclusion criterion wasestablished that minimal lesion size should be greater than 2.0 cm. From thoseinitial patients, 14 were removed due to the unavailability of survival data.

The PET machine was a dedicated whole-body PET/CT scanner (Discov-ery SDTE-16s; GE Medical Systems) in three-dimensional (3D) mode. Imageacquisition began 60 min after intravenous administration of approximately370 MBq (10 mCi) of 18F-FDG; the images obtained had a voxel size of 5.47×5.47× 3.27 mm3, with no gap between slices, and a matrix size of 128× 128.The inclusion criteria considered only newly diagnosed patients with avail-ability of pretreatment PET/CT examination and a lesion uptake higher thanbackground (SUVmax larger than twice the background), absence of distantmetastases, and a lesion size of at least 2 cm.

The 18F-FDG PET Image Analysis and Computation of theRelevant MetricsPET images in DICOM format were loaded into MATLABfor the image analysis. In each image, the tumor was manuallyselected and subsequently delineated in 3D by an automatic algo-rithm. The result of the delineation is a 3D matrix with the SUVvalues (S ) of the N voxels of the tumor. Occasionally, theremight be N ′ hypometabolic voxels enclosed by the active ones,which could be assigned to necrotic regions; these are used hereonly for geometric considerations. For all delineated tumors inPET images, we evaluated the following metrics:

MTV, i.e., volume of the delineated tumor, computed as thenumber N of selected voxels multiplied by the volume of onevoxel VV :

MTV =N ×VV . [2]

TLG, calculated as the sum of the SUV value multiplied bythe volume of the voxel for all of the N voxels in the tumor:

TLG =∑N

i=1 Si ×VV . [3]

SUVmax, maximum value of SUV in the tumor:

SUVmax = maxSi. [4]

We refer to the SUVmax voxel position by its coordinates xsm,ysm, zsm and use it as a location for the 18F-FDG hotspot.

MSR, the radius of a hypothetical sphere having the samevolume as the MTV and serving as a linear surrogate ofvolume:

MSR =

(3

4πMTV

)1/3. [5]

Centroid: As a reference point for the center of the tumorthat is independent of the observer and computable for anydelineated tumor, we use the geometrical centroid of the 3Dshape defined by the segmented tumor and all its interiorpoints, including the N ′ hypometabolic ones. Its coordinatesare computed by the mathematical definition

xc =1

N +N ′

N+N ′∑i=1

xi , [6a]

yc =1

N +N ′

N+N ′∑i=1

yi , [6b]

zc =1

N +N ′

N+N ′∑i=1

zi . [6c]

HOC, distance from 18F-FDG hotspot (voxel with SUVmax) totumor centroid measured as the Euclidean distance betweenboth points:

HOC =

√(xsm− xc)

2 + (ysm− yc)2 + (zsm− zc)

2. [7]

NHOC, normalized distance from 18F-FDG hotspot to tumorcentroid. To make the HOC size independent, we normalizeit by the MSR as a linear measure of volume. Consequently,we get a metric for the shift of the 18F-FDG hotspot that iscomparable along all tumors:

NHOC =HOC

MSR. [8]

Even though all of the images were taken with PET/CT coreg-istration, only data from positron emission tomography wereused in this work.

Kaplan–Meier Statistics. We performed Kaplan–Meier analysesover these two cohorts of patients, using the log-rank and Bres-low tests to assess the significance of the results. These methodscompare two populations separated in terms of one parameterand study their statistical differences in survival. Specifically, OSand DFS Kaplan–Meier analyses were performed. A two-tailedsignificance level with P value lower than 0.05 was applied. Thehazard ratio (HR) and its adjusted 95% confidence interval (CI)were also computed for each threshold using Cox proportionalhazards regression analysis.

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Splitting Thresholds. For each variable, we searched for everyvalue splitting the sample into two different subgroups, satisfy-ing the condition that none of them be more than five timeslarger than the other. We then tested each of them as split-ting thresholds through Kaplan–Meier analyses, obtaining thesignificance results shown in SI Appendix, Figs. S5–S8. The bestsplitting threshold was chosen as the nonisolated significant valuegiving the lowest P value in both log-rank and Breslow tests, asdescribed in ref. 52.

Harrell’s C Index. To assess the accuracy of prognostic models,Harrell’s concordance index score was also computed (60). Thismethod compares the survival of two populations of patients(best prognosis versus worst prognosis) by studying all possi-ble combinations of individuals belonging to different groups.Then, the percentage of right guesses is the reported result.Concordance indexes were computed using the noncensoredsample and ranged from 0 to 1, with 1 indicating a perfectmodel (a purely random guess would give a concordance indexof 0.5).

Variable Correlations. Spearman correlation coefficients wereused to assess the dependencies between pairs of variables.We considered significant correlation coefficients above 0.7 orbelow −0.7 as strong (direct or inverse, respectively) correlationsbetween variables. In this way we were able to exclude possibleconfounding effects in our analysis.

Statistical Software. SPSS (v. 22.0.00), MATLAB (R2018b; TheMathWorks, Inc.), and R (3.6.3) software were used for allstatistical analyses.

Nonlocal Fisher–Kolmogorov Model and Simulations. Themigration–proliferation integro-differential Eq. 1 in radialcoordinates was solved numerically using the method of lines(61) combined with Newton–Cotes integration formulas todeal with the nonlocal term. In the simulations displayedin Fig. 1, the computational domain consisted of a radialvariable r ∈ [0,Rmax], where Rmax = 7 cm was the maximumradius, and the proliferation rate ρ∈ [0, ρm], where the max-imum proliferation rate was ρm = 0.06 d−1. The number ofnodes in the discretized r -ρ mesh was 350× 180. Additionalparameters were Dc = 3.5 · 10−4 cm2/d, Dρ = 1.3 · 10−8 d−3,µ= 4 · 10−3 d−1, and K = 8 · 107 cells/cm3. The initial conditionconsisted of a highly localized lesion with a radius of 1 mm

containing 105 tumor cells and having a mean proliferation rateρ0 = 1.7 · 10−2 d−1 and standard deviation σ0 = 3 · 10−3 d−1.

The general expression for the proliferation density is

M(x, t) =

∫ ρm

0

(ρ−µ)

(1− 1

K

∫ ρm

0

u(x, ρ′, t) dρ′)u(x, ρ, t) dρ

[9]

and was used to compute the plots shown in Fig. 1B assumingspherical symmetry.

In Fig. 1C, the mean metabolic radius was defined as

Rmet(t) =

∫ Rmax

0M(r , t) r3dr∫ Rmax

0M(r , t) r2dr

, [10]

while the average proliferation rate was determined via

ρa(t) =

∫ ρm0

∫ Rmax

0ρ u(r , ρ, t) r2dr dρ∫ ρm

0

∫ Rmax

0u(r , ρ, t) r2dr dρ

. [11]

In Fig. 1D, the distance from the tumor centroid to the point ofmaximum proliferation (HOC) was calculated at each time stepvia expression 9. The NHOC was computed by means of the ratioNHOC(t) = HOC(t)/Rmet(t).

Data Availability. All study data are included in this article and/or SIAppendix. The data extracted by PET imaging processing of all of thepatients are accessible in our group’s webpage: http://matematicas.uclm.es/molab/EvolutionaryDynamicsAtTheTumorEdge DATA.xlsx.

ACKNOWLEDGMENTS. This research has been supported by grants awardedto V.M.P.-G. by the James S. Mc. Donnell Foundation, United States ofAmerica, 21st Century Science Initiative in Mathematical and ComplexSystems Approaches for Brain Cancer (collaborative award 220020560)and by the Junta de Comunidades de Castilla-La Mancha, Spain (grantnumber SBPLY/17/180501/000154). V.M.P.-G. and G.F.C. thank the fund-ing from Ministerio de Ciencia e Innovacion, Spain (grant numberPID2019-110895RB-I00). This research has also been supported by a grantawarded to G.F.C. by the Junta de Comunidades de Castilla-La Man-cha, Spain (grant number SBPLY/19/180501/000211). J.J.-S. received supportfrom Universidad de Castilla-La Mancha (grant number 2020-PREDUCLM-15634). J.J.B received support from the University of Castilla-La Man-cha (grant number 2018-CPUCLM-7798). A.M. received support fromAsociacion Pablo Ugarte (http://www.asociacionpablougarte.es). C.O.-S.received support from Asociacion Espanola Contra el Cancer (grant number2019-PRED-28372).

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